Comments on Two Theoretical Models of Enzymatic Hydrolysis of Polysaccharides
INTRODUCTION This paper points out and corrects the mathematical deficiency appearing in Refs. 1 and 2 concerning theoretical models of enzymatic hydrolysis of polysaccharides.
DISCUSSION A theoretical analysis of the degradation of polysaccharides by endo- and exoenzymes was reported by Suga et a1.l Michaelis-Menten kinetics was assumed in their derivation: k i
S
+ E-ES
k
3, P
*2
+E
'
(1)
For endoenzymes alone,
Consequently they obtained
_ dC, dt
k,CEet(i - 1)Ci + 2,:& (i - l ) C i
K,
+
k 3 C , , , Z&+l c, K , + Zy==,(i - l ) C i
(5)
In our opinion, even though eq. ( 5 ) is valid: there are certain steps in the derivation that are mathematically unjustified. By applying summation to enzyme-substrate complexes at quasi-steady-state, the following equation was obtained by Suga et al.:
From this equation, they concluded that
(CES)i
=
(i - I ) C i C E Z& (i - 1)C, + (kze; k , ) / k ,
(7)
It is evident that this conclusion is mathematically unjustifiable, and thus the subsequent derivations meaningless. Fortunately we tried and succeeded in verifying the validity of eq. (5) via a Biotechnology and Bioengineering. Vol. XXI. Pp. 236Y-237 I (1979) @ 1Y7Y John Wiley CG Sons. Inc. 0(306-35Y)2/7~/0021-2.7hY$OI .MI
2370
BIOTECHNOLOGY AND BIOENGINEERING VOL. XXI (1979)
different route. The procedure is as follows: Applying the quasi-steady-state assumption, eq. (3) becomes
-d ( C K s ) ‘- k , C E ( i- I)C, - ( k , + k 3 ) ( C , s ) i= 0 dr
or
Substitution of the above equation into eq. (4) gives
or
Substitution of eq. (9) back into eq. (8) gives
or
This equation is identical to eq. (7). Therefore through the above manipulations eq. (7) is verified and, accordingly the subsequent derivations become meaningful and eq. ( 5 ) is validated. In a later paper, Lee et al.2 extended the theoretical model to include the cellobiosylhydrolase and cellobiase activities. In developing the rate of change of C,, i = 3,4, . . . , m , for cellobiosylhydrolase, a similar derivation was incorporated. However, Lee and his coworkers did not mention any corrections and it seems that they too overlooked the mathematical deficiency in the paper by Suga et al.
Nomenclature free enzyme concentration concentration of enzyme-substrate complex with the substrate having degree of polymerization i total concentration of enzyme concentration of polysaccharide with a degree of polymerization i enzyme enzyme-substrate complex degree of polymerization of polysaccharides reaction rate constants Michaelis-Menten constant reaction product
COMMUNICATIONS TO T H E EDITOR
S f
237 I
substrate (polysaccharides) reaction time References
1. K. Suga, G. Van Dedem, and M. Moo-Young. Biorechnol. Bioeng., 17, 433 (1975). 2 . S. E. Lee, W. B. Armiger, C. M. Watteeuw, and A. E. Humphrey, Biorechnol. Bioeng., 20, 141 (1978).
T. A. Hsu* G. T. TSAO School of Chemical Engineel-ing Pui-due University West Lafayette, Indiana 47907 Accepted foi- Publication June 26, IY7Y
* To whom all correspondence should be addressed.
INTRODUCTION This paper points out and corrects the mathematical deficiency appearing in Refs. 1 and 2 concerning theoretical models of enzymatic hydrolysis of polysaccharides.
DISCUSSION A theoretical analysis of the degradation of polysaccharides by endo- and exoenzymes was reported by Suga et a1.l Michaelis-Menten kinetics was assumed in their derivation: k i
S
+ E-ES
k
3, P
*2
+E
'
(1)
For endoenzymes alone,
Consequently they obtained
_ dC, dt
k,CEet(i - 1)Ci + 2,:& (i - l ) C i
K,
+
k 3 C , , , Z&+l c, K , + Zy==,(i - l ) C i
(5)
In our opinion, even though eq. ( 5 ) is valid: there are certain steps in the derivation that are mathematically unjustified. By applying summation to enzyme-substrate complexes at quasi-steady-state, the following equation was obtained by Suga et al.:
From this equation, they concluded that
(CES)i
=
(i - I ) C i C E Z& (i - 1)C, + (kze; k , ) / k ,
(7)
It is evident that this conclusion is mathematically unjustifiable, and thus the subsequent derivations meaningless. Fortunately we tried and succeeded in verifying the validity of eq. (5) via a Biotechnology and Bioengineering. Vol. XXI. Pp. 236Y-237 I (1979) @ 1Y7Y John Wiley CG Sons. Inc. 0(306-35Y)2/7~/0021-2.7hY$OI .MI
2370
BIOTECHNOLOGY AND BIOENGINEERING VOL. XXI (1979)
different route. The procedure is as follows: Applying the quasi-steady-state assumption, eq. (3) becomes
-d ( C K s ) ‘- k , C E ( i- I)C, - ( k , + k 3 ) ( C , s ) i= 0 dr
or
Substitution of the above equation into eq. (4) gives
or
Substitution of eq. (9) back into eq. (8) gives
or
This equation is identical to eq. (7). Therefore through the above manipulations eq. (7) is verified and, accordingly the subsequent derivations become meaningful and eq. ( 5 ) is validated. In a later paper, Lee et al.2 extended the theoretical model to include the cellobiosylhydrolase and cellobiase activities. In developing the rate of change of C,, i = 3,4, . . . , m , for cellobiosylhydrolase, a similar derivation was incorporated. However, Lee and his coworkers did not mention any corrections and it seems that they too overlooked the mathematical deficiency in the paper by Suga et al.
Nomenclature free enzyme concentration concentration of enzyme-substrate complex with the substrate having degree of polymerization i total concentration of enzyme concentration of polysaccharide with a degree of polymerization i enzyme enzyme-substrate complex degree of polymerization of polysaccharides reaction rate constants Michaelis-Menten constant reaction product
COMMUNICATIONS TO T H E EDITOR
S f
237 I
substrate (polysaccharides) reaction time References
1. K. Suga, G. Van Dedem, and M. Moo-Young. Biorechnol. Bioeng., 17, 433 (1975). 2 . S. E. Lee, W. B. Armiger, C. M. Watteeuw, and A. E. Humphrey, Biorechnol. Bioeng., 20, 141 (1978).
T. A. Hsu* G. T. TSAO School of Chemical Engineel-ing Pui-due University West Lafayette, Indiana 47907 Accepted foi- Publication June 26, IY7Y
* To whom all correspondence should be addressed.
